Another criticism we note in Section REF  is that Cut models and \(\delta \) -SMI do not have correct Frequentist coverage of the pseudo-true parameters in the limit of many observations [1]}. However, first, we expect SMI to be useful when one module is well specified and we wish to bring in information from other potentially misspecified modules. In our running example, Figure REF , the Frequentist coverage of the asymptotic Cut-model posterior for \(\varphi \)  under replication of the \(Z\) -data will be correct as that module is by assumption well-specified. Secondly, in our experiments in Section REF  on a small data set the distribution of PMSE values obtained for SMI under replication of the data was not worse than Cut and Bayes and often better. This behavior is observed over a range of different levels of misspecification using fitting methods that are available in realistic settings. Finally, the cut-alternative suggested in [1]}, which does have correct asymptotic Frequentist coverage, is not an order coherent belief update.
